The counting conjecture for $S_n$-fields with local conditions

Let $K$ be an $S_n$-field for $n\geq 3$ and signature $(r_1,r_2)$. Let $\mathcal S$ be a finite set of local conditions, and let $|\mathcal S|$ denote the product of their local densities. Write $L_n^{(r_2)}(X;\mathcal S)$ for the set of such fields with $|d_K|<X$ satisfying $\mathcal S$. Counting conjecture. There are positive constants $\delta<1$ and $\gamma$ such that

$$|L_n^{(r_2)}(X)|=A(r_2)X+O(X^\delta),$$

and

$$|L_n^{(r_2)}(X;\mathcal S)|=|\mathcal S|A(r_2)X+O\left(\left(\prod_{p\in S}p\right)^\gamma X^\delta\right),$$

with the implied constant uniformly bounded over the primes $p$ and the local conditions at $p$. This conjectural uniform counting estimate is used to obtain average results for Frobenius statistics and the smallest prime in a conjugacy class; the source does not provide evidence here that it has been resolved.